Orientable Genus Research Articles

Horizontal Heegaard splittings of Seifert fibered spaces

We show that if an orientable Seifert fibered space M with an orientable genus g base space admits a strongly irreducible horizontal Heegaard splitting then there is a one-to-one correspon- dence between isotopy classes of strongly irreducible horizontal Heegaard splittings and elements of Z 2g . The correspondence is determined by the slopes of intersection of each Heegaard splitting with a collection of 2g incompressible tori in M. We also show that there are Seifert fibered spaces with infinitely many non-isotopic Heegaard splittings that determine Nielsen equivalent generating systems for the fundamental group of M.

Infinite genus surfaces and irrational polygonal billiards

We prove that the natural invariant surface associated with the billiard game on an irrational polygonal table is homeomorphic to the Loch Ness monster, that is, the only orientable infinite genus topological real surface with exactly one end.

The orientable genus of some joins of complete graphs with large edgeless graphs

In an earlier paper the authors showed that with one exception the nonorientable genus of the graph K m ¯ + K n with m ≥ n − 1 , the join of a complete graph with a large edgeless graph, is the same as the nonorientable genus of the spanning subgraph K m ¯ + K n ¯ = K m , n . The orientable genus problem for K m ¯ + K n with m ≥ n − 1 seems to be more difficult, but in this paper we find the orientable genus of some of these graphs. In particular, we determine the genus of K m ¯ + K n when n is even and m ≥ n , the genus of K m ¯ + K n when n = 2 p + 2 for p ≥ 3 and m ≥ n − 1 , and the genus of K m ¯ + K n when n = 2 p + 1 for p ≥ 3 and m ≥ n + 1 . In all of these cases the genus is the same as the genus of K m , n , namely ⌈ ( m − 2 ) ( n − 2 ) / 4 ⌉ .

On the matching extendability of graphs in surfaces

A graph G with at least 2 n + 2 vertices is said to be n-extendable if every matching of size n in G extends to a perfect matching. It is shown that (1) if a graph is embedded on a surface of Euler characteristic χ, and the number of vertices in G is large enough, the graph is not 4-extendable; (2) given g > 0 , there are infinitely many graphs of orientable genus g which are 3-extendable, and given g ¯ ⩾ 2 , there are infinitely many graphs of non-orientable genus g ¯ which are 3-extendable; and (3) if G is a 5-connected triangulation with an even number of vertices which has genus g > 0 and sufficiently large representativity, then it is 2-extendable.

Families of Dot-Product Snarks on Orientable Surfaces of Low Genus

We introduce a generalized dot product and provide some embedding conditions under which the genus of a graph does not rise under a dot product with the Petersen graph. Using these conditions, we disprove a conjecture of Tinsley and Watkins on the genus of dot products of the Petersen graph and show that both Grunbaum’s Conjecture and the Berge-Fulkerson Conjecture hold for certain infinite families of snarks. Additionally, we determine the orientable genus of four known snarks and two known snark families, construct a new example of an infinite family of snarks on the torus, and construct ten new examples of infinite families of snarks on the 2-holed torus; these last constructions allow us to show that there are genus-2 snarks of every even order n ≥ 18.

Counterexamples to the nonorientable genus conjecture for complete tripartite graphs

In 1976, Stahl and White conjectured that the minimum nonorientable genus of K l, m, n (where l≥ m≥ n) is (l−2)(m+n−2) 2 . We prove that K 4,4,1, K 4,4,3, and K 3,3,3 are counterexamples to this conjecture. We also show that all other complete tripartite graphs K l, m, n with l≥ m≥ n and l≤5 satisfy the conjecture. Moreover, all complete tripartite graphs with l≤5 satisfy the similar conjecture for orientable genus.

Blocking nonorientability of a surface

Let S be a nonorientable surface. A collection of pairwise noncrossing simple closed curves in S is a blockage if every one-sided simple closed curve in S crosses at least one of them. Robertson and Thomas [9] conjectured that the orientable genus of any graph G embedded in S with sufficiently large face-width is “roughly” equal to one-half of the minimum number of intersections of a blockage with the graph. The conjecture was disproved by Mohar (Discrete Math. 182 (1998) 245) and replaced by a similar one. In this paper, it is proved that the conjectures in Mohar (1998) and Robertson and Thomas (J. Graph Theory 15 (1991) 407) hold up to a constant error term: For any graph G embedded in S , the orientable genus of G differs from the conjectured value at most by O( g 2), where g is the genus of S .

Exponential Families of Non-isomorphic Non-triangular Orientable Genus Embeddings of Complete Graphs

It is shown that for every i∈{1,2,…,11}\\{3,4,7} the complete graph K 12 s+ i for s⩾ d( i)∈{1,2,3,4} has at least h( i)4 s non-isomorphic orientable genus embeddings, where h( i)∈{1, 1 2 , 1 4 , 1 8 }.

Topological dynamics on moduli spaces II

Let $M$ be an orientable genus $g>0$ surface with boundary $\partial M$. Let $\Gamma$ be the mapping class group of $M$ fixing $\partial M$. The group $\Gamma$ acts on ${\mathcal M}_{\mathcal C} = \operatorname {Hom}_{\mathcal C}(\pi _1(M),\operatorname {SU})/\operatorname {SU},$ the space of $\operatorname {SU}$-gauge equivalence classes of flat $\operatorname {SU}$-connections on $M$ with fixed holonomy on $\partial M$. We study the topological dynamics of the $\Gamma$-action and give conditions for the individual $\Gamma$-orbits to be dense in ${\mathcal M}_{\mathcal C}$.

Topological dynamics on moduli spaces, I

Let M be an orientable genus g > 0 surface with boundary O3M. Let 17 be the mapping class group of M fixing OM. The group 17 acts on Mc = Homc(7r1(M),SU(2))/SU(2), the space of SU(2)-gauge equivalence classes of flat SU(2)-connections on M with fixed holonomy on O3M. We study the topological dynamics of the 1-action and give conditions for the individual F-orbits to be dense in Mc.

Restrictions of representations of a surface group to a pair of free subgroups

Let Π be the fundamental group of a compact orientable genus m surface, and let G be a connected reductive algebraic group over an algebraically closed field of characteristic zero. Define two free rank m subgroups of Π by A = 〈a1, …, am〉 and B = 〈b1, …, bm〉, where Π = 〈a1, …, am, b1, …, bm |Πj=1m[aj, bj]〉 is the standard presentation of Π. We consider representations of Π, of A and of B into G. Restriction of representations induces a morphism from C(Π, G), the variety of closed conjugacy classes of representations of Π, to C(A, G)C(B, G). We prove that if m is greater than the semisimple rank of G then this morphism is dominant and almost all fibers are finite.

Sharp lower bounds of the least eigenvalue of planar graphs

Let G be a simple graph with n⩾3 vertices and orientable genus g and non-orientable genus h. We define the Euler characteristic χ( G) of a graph G by χ( G)=max{2−2 g,2− h}. Let λ( G) be the least eigenvalue of the adjacency matrix A of G. In this paper, we obtain the following lower bounds of λ( G) λ(G)⩾− 2(n−χ(G)) . In particular, if G is the planar graph, then λ(G)⩾− 2n−4 the equality holds if and only if G≅ K 2, n−2 . Further, we have same result of series–parallel graph.

On the Connectivity of Graphs Embedded in Surfaces

In a 1973 paper, Cooke obtained an upper bound on the possible connectivity of a graph embedded in a surface (orientable or nonorientable) of fixed genus. Furthermore, he claimed that for each orientable genus γ >0 (respectively, nonorientable genus γ >0, γ ≠2) there is a complete graph of orientable genus γ (respectively, nonorientable genus γ ) and having connectivity attaining his bound. It is false that there is a complete graph of genus γ (respectively, nonorientable genus γ ), for every γ (respectively γ ) and that is the starting point of the present paper. Ringel and Youngs did show that for each γ >0 (respectively, γ >0, γ ≠2) there is a complete graph K n which embeds in S γ (respectively N γ ) such that n is the chromatic number of surface S γ (respectively, the chromatic number of surface N γ ). One then easily observes that the connectivity of this K n attains the upper bound found by Cook. This leads us to define two kinds of connectivity bound for each orientable (or nonorientable) surface. We define the maximum connectivity κ max of the orientable surface S γ to be the maximum connectivity of any graph embeddable in the surface and the genus connectivity κ gen ( S γ ) of the surface to be the maximum connectivity of any graph which genus embeds in the surface. For nonorientable surfaces, the bounds κ max ( N γ ) and κ gen ( N γ ) are defined similarly. In this paper we first study the uniqueness of graphs possessing connectivity κ max ( S γ ) or κ max ( N γ ). The remainder of the paper is devoted to the study of the spectrum of values of genera in the intervals [ γ ( K n )+1, γ ( K n +1 )] and [ γ ( K n )+1, γ ( K n +1 )] with respect to their genus and maximum connectivities.

On the orientable genus of graphs with bounded nonorientable genus

A conjecture of Robertson and Thomas on the orientable genus of graphs with a given nonorientable embedding is disproved.

Embeddings of one kind of graphs

The purpose of this paper is to display a new kind of simple graphs which belong to B n in which any graph has its orientable genus n,n≥3. Furthermore, for any integer k,1≥ -k≤ n, there exists a graph B n k of B n such that the non-orientable genus of B n k is k.

On the minimal genus of 2-complexes

Gross and Rosen asked if the genus of a 2-dimensional complex K embeddable in some (orientable) surface is equal to the genus of the graph of appropriate barycentric subdivision of K. We answer the nonorientable genus and the Euler genus versions of Gross and Rosen's question in affirmative. We show that this is not the case for the orientable genus by proving that taking ⌊ log2 g⌋ th barycentric subdivision is not sufficient, where g is the genus of K. On the other hand, (1+⌈log2(g+2)⌉)th subdivision is proved to be sufficient. © 1997 John Wiley & Sons, Inc.

The Genus of the Coxeter Graph

AbstractIn [1], Biggs stated that the Coxeter graph can be embedded in an orientable surface of genus 3. The purpose of this note is to point out that Biggs' embedding is in fact into a non-orientable surface. Further, it is shown that the orientable genus is 3 and the non-orientable genus is 6.

Computing the orientable genus of projective graphs

AbstractThe orientable genus is determined for any graph that embeds into the projective plane, Σ, to be essentially half of the representativity of any embedding into Σ. In addition, a structure is given for any 3‐connected projective planar graph as the union of a spanning planar graph and a variation of a Möbius Ladder. © 1995 John Wiley & Sons, Inc.

Bounds for the average genus of the vertex-amalgamation of graphs

The average orientable genus of graphs has been the subject of a considerable number of recent investigations. It is the purpose of this article to examine the extent to which the average genus of the amalgamation of graphs fails to be additive over its constituent subgraphs. This discrepancy is bounded and the sharpness of these bounds discussed and compared to similar bounds for the minimum and the maximum genera of the amalgamation of graphs. The main result relies on permutation-partition pairs for its proof.

The genus of a random graph

The orientable genus of a graph is the minimum number of handles needed to embed that graph on a surface. Determining the genus of a graph is a fundamental yet very difficult problem. In this paper we show that the orientable genus of a randomly selected graph on n vertices has a strong tendency towards n 2/24. More strongly, we give upper and lower bounds on the genus which hold for almost all graphs. We extend our results to the graph model in which the edge-probability varies with the order of the graph and to the model in which the number of edges is preordained. We also give the analogous bounds for the nonorientable genus. Loosely speaking, a random graph almost always has embeddings on both orientable and nonorientable surfaces which are nearly triangulations.